Describe the end behavior of each function.
1. P(x) = 2x^3 + x^2 + 3x + 2
2. P(x) = 5x^4 - 6x^6 + 3x + 2
If by "end behavior" you mean the behavior for very large positive or negative x, then 1. behaves like 2x^3 and 2. behaves like -6x^6. The highest power of x always dominates when x becomes very large in either the positive or negative direction.
To determine the end behavior of a function, we need to observe the power (degree) and the coefficient of the leading term of the polynomial.
For the first function, P(x) = 2x^3 + x^2 + 3x + 2, the leading term is 2x^3. The degree of the polynomial is determined by the highest power of x, which in this case is 3. The coefficient of the leading term is 2.
Since the degree is odd and the leading coefficient is positive, we can conclude that as x approaches positive or negative infinity, the function P(x) will also approach positive or negative infinity, respectively. This means that the end behavior of P(x) is an upward trend.
Now, let's analyze the second function, P(x) = 5x^4 - 6x^6 + 3x + 2. In this case, the leading term is -6x^6, with a degree of 6. The coefficient of the leading term is -6.
Since the degree is even and the leading coefficient is negative, as x approaches positive or negative infinity, the function P(x) will approach negative infinity. Therefore, the end behavior of P(x) is a downward trend.
In summary:
1. The end behavior of P(x) = 2x^3 + x^2 + 3x + 2 is a positive (upward) trend as x approaches positive or negative infinity.
2. The end behavior of P(x) = 5x^4 - 6x^6 + 3x + 2 is a negative (downward) trend as x approaches positive or negative infinity.